Problem: $g'(x)=8g(x)$, and $g(2)=7$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $g(x)=7e^{2x+8}$ (Choice B) B $g(x)=7e^{8x-16}$ (Choice C) C $g(x)=8e^{7x+2}$ (Choice D) D $g(x)=e^{8x}+7$
Answer: The general solution of equations of the form $g'(x)=kg(x)$ is $g(x)=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=8$, so $g(x)=C\cdot e^{8x}$. Let's use the fact that $g(2)=7$ to find $C$ : $\begin{aligned} g(x)&=C\cdot e^{8x} \\\\ g(2)&=C\cdot e^{8\cdot 2} \gray{\text{Plug }x=2} \\\\ 7&=C\cdot e^{8\cdot 2} \gray{g(2)=7} \\\\ 7e^{-16}&=C \end{aligned}$ In conclusion, $g(x)=7e^{8x-16}$.